arXiv subjects

Yuehui Zhang

Publications and source records attributed to Yuehui Zhang.

9 recordsLinked to original sources

arXiv2026

Directed partial orders on the complex number field

We construct a class of positive cones that make $\mathbb{C}$ into a directed partially ordered ring. The positive cones are defined using integral closures of local rings, associated with a transcendence basis and a chosen real generator. A localization criterion also yields such orders on every transcendental extension of $\mathbb{Q}$. For a fixed coefficient field, we prove that two real generators define the same cone if and only if they differ by an affine map with positive real-algebraic slope and translation algebraic over that field. The nonzero positive elements are closed under inversion.

math.RA↗

arXiv2026

A lattice order on the complex field

Birkhoff and Pierce asked in 1956 whether the field $\mathbb{C}$ of complex numbers admits a lattice order compatible with addition and multiplication. We answer the question affirmatively. Using a valuation with value group $\mathbb{Q}$, we construct a positive cone $P$ by a transfinite recursion of countable ordinal length. At each stage we extend a lattice order on a subspace while preserving its lattice operations in the whole field and maintaining multiplicative closure of the ambient positive cone. These subspaces eventually exhaust $\mathbb{C}$. The construction depends on choices of a valuation extension, compatible rational powers and vector-space bases. The cone $P$ contains $1$ and satisfies $\{x\in P\setminus\{0\}:x^{-1}\in P\}=\mathbb{Q}_{>0}$. The induced order is not total: $i$ is incomparable with $0$. For a chosen transcendental element $t$ and $i^2=-1$, one may prescribe $0\vee i=(t+i)/2$ and $0\vee(-i)=(t-i)/2$. Varying the first extension yields exactly $2^{\mathbf c}$ distinct lattice positive cones on $\mathbb{C}$, where $\mathbf c=|\mathbb{C}|$. The count concerns cones as subsets of $\mathbb{C}$.

math.RA↗

arXiv2026

Quasi-Hereditary Orderings of Nakayama Algebras

Let $A$ be an algebra, and let $\mathcal{S}$ be the set of isomorphism classes of simple $A$-modules, with $|\mathcal{S}|=n$. In this paper, a total ordering of $\mathcal{S}$ is called a quasi-hereditary ordering, or a $q$-ordering, for $A$ if every Weyl module is Schurian and every indecomposable projective $A$-module is filtered by Weyl modules. The number of such total orderings is denoted by $q(A)$. Determining whether a given total ordering of $\mathcal{S}$ is a $q$-ordering is a difficult problem. A result of Dlab and Ringel states that $A$ is hereditary if and only if every total ordering of $\mathcal{S}$ is a $q$-ordering; equivalently, $q(A)=n!$. The $q$-ordering conjecture, proposed in 2000, asserts that every non-hereditary algebra $A$ satisfies $q(A)\le\dfrac{2}{3}n!$. The main result of this paper is a necessary and sufficient criterion for a total ordering of the simple modules of a Nakayama algebra to be a $q$-ordering. More precisely, we show that a total ordering is a $q$-ordering if and only if, for every minimal relation $g$, the maximal element of $\operatorname{Hod}(g)$ does not belong to $\operatorname{Int}(g)$. As consequences, we characterize quasi-hereditary Nakayama algebras by the non-emptiness of the $X$-set, derive an iteration formula for $q(A)$, and prove the $q$-ordering conjecture for Nakayama algebras. Finally, we give a non-monomial example showing that the conjecture does not hold for general finite-dimensional algebras.

math.RT↗

arXiv2021

Nilpotent Category of Abelian Category and Self-Adjoint Functors

Let $\mathcal{C}$ be an additive category. The nilpotent category $\mathrm{Nil} (\mathcal{C})$ of $\mathcal{C}$, consists of objects pairs $(X, x)$ with $X\in\mathcal{C}, x\in\mathrm{End}_{\mathcal{C}}(X)$ such that $x^n=0$ for some positive integer $n$, and a morphism $f:(X, x)\rightarrow (Y,y)$ is $f\in \mathrm{Hom}_{\mathcal{C}}(X, Y)$ satisfying $fx=yf$. A general theory of $\mathrm{Nil}(\mathcal{C})$ is established and it is abelian in the case that $\mathcal{C}$ is abelian. Two abelian categories are equivalent if and only if their nilpotent categories are equivalent, which generalizes a Song, Wu, and Zhang's result. As an application, it is proved all self-adjoint functors are naturally isomorphic to $\mathrm{Hom}$ and $\mathrm{Tensor}$ functors over the category $\mathrm{Nil}$ of finite-dimensional vector spaces. Both $\mathrm{Hom}$ and $\mathrm{Tensor}$ can be naturally generalized to $\mathrm{HOM}$ and $\mathrm{Tensor}$ functor over $\mathrm{Nil}(\mathcal{V})$. They are still self-adjoint, but intrinsically different.

math.CT↗

arXiv2021

A Generalized Determinant of Matrices and Applications

A generalized definition of the determinant of matrices is given, which is compatible with the usual determinant for square matrices and keeps many important properties, such as being an alternating multilinear function, keeping multiplication formula and partly keeping the Cauchy-Binet's formula. As applications of the new theory, the generalized Cramer's rule and the generalized oriented volume are obtained.

math.CA↗

arXiv2016

Unbounded ladders induced by Gorenstein algebras

The derived category $D({\rm Mod}A)$ of a Gorenstein triangular matrix algebra $A$ admits an unbounded ladder; and this ladder restricts to $D^-({\rm Mod})$ {\rm(}resp. $D^b({\rm Mod})$, $D^b({\rm mod})$, $K^b({\rm proj})${\rm)}. A left recollement of triangulated categories with Serre functors sits in a ladder of period $1$; as an application, the singularity category of $A$ admits a ladder of period $1$.

math.RT↗

arXiv2016

Unbounded ladders induced by Gorenstein algebras

The derived category of a Gorenstein triangular matrix algebra $A$ admits an unbounded ladder, which is of period $3$ if $A = T_2(B)$. Also, a left recollement of triangulated categories with Serre functors sits in a ladder of period $1$; as an application, the singularity category of $A$ admits a ladder of period $1$.

math.RT↗

arXiv2013

Injective Objects of Monomorphism Categories

For an acyclic quiver $Q$ and a finite-dimensional algebra $A$, we give a unified form of the indecomposable injective objects in the monomorphism category ${\rm Mon}(Q,A)$ and prove that ${\rm Mon}(Q, A)$ has enough injective objects. As applications, we show that for a given self-injective algebra $A$, a tilting object in the stable category $\underline{A}$-mod induces a natural tilting object in the stable monomorphism category $\underline{\rm Mon}(Q,A)$. We also realize the singularity category of the algebra $kQ\otimes_k A$ as the stable monomorphism category of the module category of $A$.

math.RT↗

arXiv2013

Repetitive cluster-tilted algebras

Let $H$ be a finite dimensional hereditary algebra over an algebraically closed field $k$ and $\mathscr{C}_{F^m}$ be the repetitive cluster category of $H$ with $m\geq 1$. We investigate the properties of cluster tilting objects in $\mathscr{C}_{F^m}$ and the structure of repetitive cluster-tilted algebras. Moreover, we generalized Theorem 4.2 in \cite{bmrrt} (Buan A, Marsh R, Reiten I. Cluster-tilted algebra. Trans. Amer. Math. Soc., 359(1)(2007), 323-332.) to the situation of $\mathscr{C}_{F^m}$, and prove that the tilting graph $\mathscr{K}_{\mathscr{C}_{F^m}}$ of $\mathscr{C}_{F^m}$ is connected.

math.RT↗